**Algebraic Number Theory and Quadratic Fields**

Algebraic Number Fields

The Gaussian Field

Euclidean Quadratic Fields

Applications of Unique Factorization

**Ideals **

The Arithmetic of Ideals in Quadratic Fields

Dedekind Domains

Application to Factoring

**Binary Quadratic Forms**

Basics

Composition and the Form Class Group

Applications via Ambiguity

Genus

Representation

Equivalence Modulo *p *

**Diophantine Approximation**

Algebraic and Transcendental Numbers

Transcendence

Minkowski’s Convex Body Theorem

**Arithmetic Functions**

The Euler–Maclaurin Summation Formula

Average Orders

The Riemann *zeta*-function

**Introduction to ***p*-Adic Analysis

Solving Modulo *p*^{n}

Introduction to Valuations

Non-Archimedean vs. Archimedean Valuations

Representation of *p*-Adic Numbers

**Dirichlet: Characters, Density, and Primes in Progression**

Dirichlet Characters

Dirichlet’s *L*-Function and Theorem

Dirichlet Density

**Applications to Diophantine Equations**

Lucas–Lehmer Theory

Generalized Ramanujan–Nagell Equations

Bachet’s Equation

The Fermat Equation

Catalan and the ABC-Conjecture

**Elliptic Curves**

The Basics

Mazur, Siegel, and Reduction

Applications: Factoring and Primality Testing

Elliptic Curve Cryptography (ECC)

**Modular Forms**

The Modular Group

Modular Forms and Functions

Applications to Elliptic Curves

Shimura–Taniyama–Weil and FLT

**Appendix: Sieve Methods **

**Bibliography **

**Solutions to Odd-Numbered Exercises **

**Index: List of Symbols**

**Index: Alphabetical Listing**